1990 AMC 12 第 6 题

Question

点

A

与

B

相距

5

个单位。在一个给定的、包含

A

和

B

的平面内，有多少条直线到这两点的距离分别为

2

个单位（到

A

）和

3

个单位（到

B

）？

Points

A

and

B

are

5

units apart. How many lines in a given plane containing

A

and

B

are

2

units from

A

and

3

units from

B?

0

1

2

3

多于

3

条

more than

3

答案：D

知识点：distance from point to line 切线圆

难度评级：1500

小提示：

将距离条件转化为两圆的切线条件

Replace the distance conditions by tangencies to two circles

大提示：

两圆的半径分别为

2

和

3

，且两圆外切

The circles have radii

2

and

3

and are externally tangent

解答：

符合条件的直线，是以

A

为圆心、半径为

2

的圆与以

B

为圆心、半径为

3

的圆的公切线。两圆的圆心距等于半径之和，所以两圆外切。它们有两条外公切线，并在切点处另有一条公切线，共有

3

条。

所以正确答案是 D。

A qualifying line is a common tangent to the circle centered at

A

with radius

2

and the circle centered at

B

with radius

3.

Their center distance equals the sum of their radii, so they are externally tangent. They have two external common tangents and one common tangent at their point of contact, for a total of

3.

Thus the correct answer is D.

1990 AMC 12 第 6 题

答案：D

难度评级：1500

小提示：

大提示：

解答：

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